Above all: Motivation – some parts of our Maths for Engineers teaching concept

We present some parts of our Maths for Engineers teaching concept: above all, motivation: sometimes mathematics teachers observe that engineering students, for example, perceive the maths concepts as a useless evil for them and therefore learn with little motivation and sustainability. When the maths concepts are later used in other subjects, the students do not know how to apply them and teachers do not value the maths education very highly. We solve the motivation problem by linking the mathematics with the subject education: the maths exercise program contains tasks from the subject education at that moment when the necessary maths concepts are available; examples can be found here: https://tud.link/y33m. In this contribution we will document our experience in achieving this maths‐subject‐link and the success by motivating students.


MOTIVATION
We set the section-counter to zero in order to emphasize that motivation comes before everything else.To exaggerate a bit: if you do not motivate your students you can leave out 1., 2., . . .too.Thus it would be a contradiction in terms if we would start this contribution without motivation: we receive persistent praise from our students for our teaching, s. for example, Table 1.So we think it is possibly of interest for others to get to know some parts of our teaching concept.
TA B L E 1 Statements of students of our Maths for Engineers-course made in a survey in May 2023.

Translation (Engl.)
Maths was definitely the best modul during the first two years concerning the organization.
Best maths-course I ever had.
Super lecture and tutorials.So the topics are very understandable and one has no fear to fail in the exam.
This paper is organized as follows: in Section 1 we explain how we solve the motivation problem for our students.We describe our hybrid course design in Section 2 and the design of the tutorials in Section 3. Finally we present two ways how to bribe students with extra-points for the exam in Section 4.
Throughout the paper we present results of a survey performed in May 2023.The cohort consisted of our students who enrolled in electrical engineering in 2021.Thus at that moment they had enjoyed our maths-course already for nearly four terms.

MOTIVATION BY MEANS OF LINKING THE MATHS-COURSE WITH FOLLOW-UP APPLIED-LECTURES
We are convinced that the lecturer is responsible for the motivation of his students.In the sense that he makes clear why it makes sense for students to learn what he wants to teach, thus creating a motivating learning environment.In general only the lecturer has the overview, what the things he teaches can be useful for.It is difficult for students to have such an overview.
Thus we as Maths for Engineers lecturers link our mathematical contents with follow-up lectures, where the mathematical skills will be of use for the students.We observe that the students are much more eager to learn the mathematical concepts if they can apply them either immediately or have at least the vision to apply it later in a different subject.
This link to other subjects is realized in two ways: • Within the lecture: we explicitly name the applications, s. Figure 1 in order to get an impression of named applications in this case of complex numbers.• Within the tutorials: the maths exercise program contains application tasks at that moment when the necessary maths concepts are available.We either create application tasks ourselves or use tasks out of the exercise program of followup subjects.This motivates the students and, at the same time, they enjoy it because they are convinced that they are learning something really useful.See Figures 2 and 3 for two examples of application tasks we gave our students within the exercise program of Maths 3 -complex analysis.More examples can be found here: https://tud.link/y33m.
You can imagine that quite some knowledge of the follow-up subjects is necessary in order to be able to create such detailed application tasks.When we asked the lecturers of the follow-up subjects for such application tasks, the response was basically zero.So we inspected the follow-up lectures ourselves.This was easy for us, as the first author taught quite some of the follow-up lectures himself.But it is worth doing so for every maths lecturer -for the sake of motivation of his students.

HYBRID COURSE DESIGN: USE OF RECORDINGS OF LECTURES AND TUTORIALS
Our course design is hybrid in the following sense: we do face-to-face lectures and tutorials but offer recordings of the lectures and of the central-tutorials, s.Section 3, too.See Figure 4 for an example course plan with linked live-notes (handwritten script of the lecture), slides and video-records.The original can be found here: https://tud.link/r9e7.Thus the students can decide whether to attend the face-to-face lecture or watch the video at home in a 'flipped-classroom-style' 1 .4/5 of our students choose face-to-face 2 but use the recordings for review and exam preparation.
A side-effect of making script and recordings of lectures in such a nice overview like in Figure 4 available is that now follow-up lecturers have easy access to the contents of the maths lectures.So they can refer to the things which where already taught and can start exactly there where the maths course stopped.This hopefully helps to avoid unnecessary repetition.
1 Roughly speaking 'flipped classroom learning' can be seen as consisting of two parts: 1st.acquire knowhow left alone with material in asynchronous form, 2nd. the so called quality-teaching time with direct exchange between teacher and learner in face-to-face, basically the conventional tutorial. 2This seems to indicate that (at least in our case) despite all alleged advantages 'flipped-classroom' is said to have, that the students if they have the choice prefer to replace the first part of the 'flipped-classroom' by face-to-face.This of course results in conventional teaching: lecture and tutorial in a face-to-face setting.
F I G U R E 3 Application task within the maths exercises with link to the lecture 'Dynamische Netzwerke'.

TUTORIAL DESIGN: CENTRAL-TUTORIAL PLUS SMALL-GROUP-TUTORIAL
We changed our tutorial design in the following way in 2021: instead of former TWO small-group-tutorials, we now offer ONE central-tutorial and ONE small-group-tutorial per week.The students generally welcome the new design.See Figure 5 for the students point of view on that and find some comments given by the students concerning the central-tutorial in Table 2.The success of our changed tutorial design is also documented by changes in exam scores: the pass rate did not change, but we are observing a raised midfield, see Figure 6.We have to admit that the success of the new tutorial design relies on at least two things: • The central-tutorial has to be given by a really experienced lecturer, who knows the difficulties of the students very well and who addresses even a big audience very good and • a well-thought-out choice of which exercises are due in the central-tutorial and which in the following small-grouptutorial.It is by no means just splitting the exercises of the former two small-group-tutorials in two parts.Instead one has to choose carefully what is best demonstrated and explained to the students in the central-tutorial such that the students are able so solve the remaining tasks alone.

BRIBERY WITH EXTRA-POINTS FOR THE EXAM BY MEANS OF SPECIAL RULES
In the following we describe how we try to 1. keep the students continuously 'staying on task' during the term, TA B L E 2 Statements of students concerning the central-tutorial made in a survey in May 2023.

Translation (Engl.)
1 central-tutorial + 1 small-group-tutorial, as the central-tutorial was recorded.By help of these records one could better solve the exercises of the small-group-tutorial.
The central-tutorial often provided better understanding.
The central-tutorials provided a good overview on the central maths concepts.The records were useful for repetition.2. bring students into learning-groups and 3. keep the students reviewing what they have learned so far from time to time during the term.
We do this by means of extra-credit rules the German version of which can be found here: https://tud.link/6r84.

Tasks to be submitted weekly but only sometimes marked
Every week we choose one task the students can prepare to submit.Random-number-generator-based is decided which week the prepared task is marked.So we mark only three to four times per term a task and in an easy 'caus binary way: one point or none.But this way we achieve that the students prepare at least one task basically every week and they like it, s. Figure 7A.

Tasks to be performed in learning-groups
The second part of extra-points can be achieved only if the student joines, that is, subscribes, a learning-group of three to five persons.This learning-group has to prepare for each finished chapter of the lecture a one-page-summary, s. Figure 8 for an example.These chapter-summaries have to be submitted by several deadlines during the term and are checked by the tutor.If all chapter-summaries of the learning-group are fine the students get two extra-points for the exam.This is virtually nothing compared to the 100 points of the exam but the students like it, s. Figure 7B and Table 3.

F I G U R E 8
Example chapter-summary on integral-calculus of functions with more than one variable by the learning-group of Nicolas Bachmann.

CONCLUSION
We presented some parts of our Maths for Engineers teaching concept.On the one hand, they improve the motivation of the students, which hopefully means that the knowledge is much more sustainable and the students do not forget everything a few weeks after the exam.On the other hand, they imply that a large part of the students acquire knowledge better than before.
We wish others success in perhaps trying to realize some parts of our concept.

Translation (Engl.)
We partly found through the chapter-summaries into a bigger learning-group.Finally it emerged out of 3 learning-groups.At the end it was very useful to do this in learning-groups.
The idea of the chapter-summaries is very successful, especially as we produced several versions in order to choose eventually the best.

A C K N O W L E D G M E N T S
Open access funding enabled and organized by Projekt DEAL.

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I G U R E 1 Applications of complex numbers named within the maths-lecture; painting by Simon Puteanus, Fak.EuI, TU Dresden.F I G U R E 2Application task within the maths exercises with link to control theory.

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Course plan with linked live-notes (handwritten of the lecture), slides and video-records, s. https://tud.link/r9e7.F I G U R E 5 Students point of view on what they prefer (survey in May 2023): A = 1 central-tutorial + 1 small-group-tutorial, B = no preference, C = 2 small-group-tutorials.

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Change in exam scores: (left) before and (right) after the introduction of the central-tutorial in 2021: unchanged pass rate, but raised midfield (100 = very good, 500 = failed).

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Students response on the questions (a) Do you consider the chance to get extra-points by weekly prepared tasks useful?(N = 61) and (b) Do you consider the chance to get extra-points by chapter-summaries prepared by your learning-group useful?(N = 60) A = useful, B = neutral, C = not useful (survey in May 2023).